Ok guys, so here is an update. And a quite a weird one.

I listened to your suggestions and checked the furnace and tried to
figure this one out.
I tried to see if it's dirty inside, and it is, dirty. Not entirely
hideous, but 'hey' it's 20 years old and I doubt it was ever cleaned.
I can't post the picture here, so just imagine, that the level of
dirtiness inclines more to the negative side, if you know what I
mean

So, it's not very clean, dust everywhere inside, but I doubt that it
is less dirty than it was this March (when it was still running).

In my opinion this has nothing to do with how dirty it is.

Then I checked the electric box and as Edwin Pawlowski said in this
thread, voltage is low. It's not enough, only 14volts... why?
I've no clue, because I don't have much knowledge of the topic.

In the mean time, pilot is still working, everything else seems ok,
except the burned never goes on. And that is probably caused by not
enough voltage, thus gas valve is not opening.

Anyhow, so because I know sh*t about what to do, I called an
electrician/furnace repair man.

This guy walks into our basement, kneels next to the furnace, doesn't
do anything just looks at the burning pilot, shakes his head and says,
that we should immediately turn it off, if we don't want to die.
I am looking at him, trying to figure out, how the hell he could find
the problem so quickly without checking anything.
So I asked him if he could check the electric box and the voltage. He
said, that gas valve is not controlled by it and we need to replace
the whole unit immediately and if I don't have the money, I should get
a loan or financing.
I repeated if he can take a look at the box, that there is not enough
voltage and that from my point of view, if there are going electric
cables from the box to the valve box, it is probably controlling the
valve.

This got him really upset, but he checked it and said, that there is a
problem with the box, it has only 14volts
So, I asked if the transformer is not working. He said: most probably
not.

So, he was a bit shaking at this point and told me that it's old,
dirty. That cleaning will cost me $600 and the new unit $2000.
That he won't repair electric box, because even if it worked, he
doesn't want to have anything to do with the fact it will kill us
afterwards... and so on and on... wouldn't let me say one word.

When he eventually stopped, I asked if he can clearly tell what the
problem is. He said: electric box and that it's dirty.
I said: I will clean it up, you repair the box.
He said: No, got ****ed, closed the gas going to our furnace and left
totally ****ed. I was walking right behind him, asking: "And, how much
do we owe you?"
He replied: "Nothing, if something happens here and you died, cheque/
bill would be a proof I was here and they could put me to jail!

Ufff... so he gave me a scare at the point where he would accept any
cash, until then he seems only like a regular asshole, trying to sell
me a unit without even wanting to take a look at what the problem is.

Can someone tell me what my next step should be?
Should I call another electrician and let him repair the transformer
from 120 to 24volt and run the unit, or forget it and buy a new
furnace?

I am kind of worried now, because of what he said and he is definitely
a bigger expert than I am, but I have to think logically too...

What do you suggest?